Question

Two identical metal balls one at T₁ = 300 K and the other at T₂ = 600 K are kept at a distance of 1 m in a vacuum. Will the temperatures equalise by radiation? Will the rate of heat gained by the colder sphere be proportional to ${\mathrm{T}}_2^4-{\mathrm{T}}_1^4$ as may be expected from the Stefan's law?

Answer 1

Yes, the temperature of the balls can be equalised by radiation. This is because both the spheres will emit radiations in all the directions at different rates.
The ball kept at the temperature of 300 K will gain some thermal energy by the radiation emitted by the ball kept at the temperature of 600 K. Also, it losses energy by radiation.
Similarly, the ball kept at the temperature of 600 K will gain some thermal energy by the radiation emitted by the ball kept at the temperature of 600 K. Also, it losses energy by radiation.
A time comes when the temperature of both the bodies becomes equal.

Yes, the rate of heat gained by the colder sphere is proportional to $T_2^4-T_1^4$.